Metrizability of the space of quasicontinuous functions (Q725663)

For a Hausdorff space \(X\) and a metric space \(Y\) the symbol \(Q(X,Y)\) denotes the set of all quasi-continuous (known also as semi-continuous) mappings from \(X\) to \(Y\). The authors study several countability conditions of the topological space \(Q(X,Y)\) equipped with the topology \(\tau_{uc}\) of uniform convergence on compacta and the uniform space \((Q(X,Y),\mathcal U_{uc})\). It is shown, for example, that metrizability and first countability are equivalent in the space \((Q(X,Y),\tau_{uc})\) and are equivalent to hemicompactness of \(X\). Sequentially compact subsets of \((Q(X,Y),\tau_{uc})\) are characterized.